On Nassau there's really only two options remaining. Stuart Cove's or Bahamas Divers. BD is at the base of the bridge to Paradise Island, SC's is on the south side of New Providence and does daily shuttles over and back from Paradise Island.
So that's got to cut into your dive time if you want to dive a lot. Afaik from reading here, there's no shore diving on New Providence - didn't notice anyone doing it while there either. There's also a water temperature drop in the Bahamas during the winter months if that's a consideration.
I've read many comments indicating that SC is not what you're looking for, they have 7 boats - up to 24 divers per boat, they limit bottom time and get most of the cruise ship trade. Most of the posts I read praising them are from fairly new or cruise ship divers and a lot deal with their shark feed - there are a lot of posts about SC in this forum so read some and decide for yourself. Not for me...
Although I've recently read somewhere that BD also tries to limit bottom time to make their schedule. Might be good to e-mail both to inquire.
One thing about diving in that area of the Caribbean is that at times the water tends to not be so clear as say Maui. But other times it's great. Just depends when/where you go.
It's very different diving in that area of the Caribbean than where you've been so far, the substrate is sand instead of rocky formations and the corals literally grow on top of each other in places. Add in the soft corals, fans etc. and it's a completely diifferent dive experience. I've dove Hawaii many times, the Sea of Cortez once and multiple Caribbean destinations for comparison.
Other places to consider that would meet both your/family's needs:
As mentioned - Grand Cayman - staying on Seven Mile Beach to meet the beach/tanning requirement. The diveops will pick you up either by boat or shuttle van anywhere along there for the short ride to their boats. There's several great easy shoredives both north and south of there - no diving off SMB. Clearest water you've ever seen also. A lot of the first dives off the boats tend to be deep along/down the walls so maybe get your AOW before you go. But whoooo is it pricey...makes the Bahamas look cheap...
Turks/Caicos has great beaches - especially near Grace Bay but it's all boat diving. About all I know about it - except it's on the short list for future dive trips.
St. Martin/St. Maarten might be another consideration, especially a small villa on the beach, you can do some shorediving in certain areas, and also dive Antigua and Saba on day trips. If we could've worked out the logistics, my group was going there next month.
) - would there be enough quality dive sites to explore on Nassau/Paradise Island? Any shore diving in that area? 
